Answer to Question #250702 in Financial Math for Bileni

Question

Starting in 7 years and 9 months you want to be able to withdraw $1700 at the beginning of every month. You deposit $100 000.00 immediately and then let it grow at a rate of 6.41% compounded quarterly. For how many years will you be able to withdraw these payments?

Answer 1

To find compound interest we use:

$A=P(1+\frac{r}{n})^{nt}$

Principal$=$ 100,000.00

r$=$ $\frac{6.41}{400}=0.016025$

nt$=$ 31 quarters

$A=100000(1+0.016025)^{31}$

$A=163,695.3037$

For Payout annuity we use the formula

$P_0= \frac{PMT(1-(1+\frac{r}{n})^{-nt}}{\frac{r}{n}}$

$P_0=$ Account balance at the beginning(163,695.30)

PMT$=$ Regular withdrawal amount(1700)

r$=$ Annual interest rate(decimal)(0.0641)

n$=$ Number of compounds per year(12)

t$=$ Number of years to withdraw(?)

$163,695.30= \frac{1700(1-(1+\frac{0.0641}{12})^{-12t}}{\frac{0.0641}{12}}$

$\frac{163,695.30×0.0053}{1700}=1-1.0053^{-12t}$

$0.514356=1-1.0053^{-12t}$

$1.0053^{-12t}=1-0.514356$

$log 1.0053^{-12t}=log 0.485644$

$-12t=\frac{log0.485644}{log1.0053}$

$-12t=136.64$

$t= 11.38 years$